| L(s) = 1 | − 2-s + 4-s − 8-s + 2·13-s + 16-s + 6·17-s − 8·19-s − 2·26-s − 6·29-s + 4·31-s − 32-s − 6·34-s + 10·37-s + 8·38-s − 6·41-s + 4·43-s + 2·52-s − 6·53-s + 6·58-s − 12·59-s + 10·61-s − 4·62-s + 64-s + 4·67-s + 6·68-s − 12·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.554·13-s + 1/4·16-s + 1.45·17-s − 1.83·19-s − 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 1.29·38-s − 0.937·41-s + 0.609·43-s + 0.277·52-s − 0.824·53-s + 0.787·58-s − 1.56·59-s + 1.28·61-s − 0.508·62-s + 1/8·64-s + 0.488·67-s + 0.727·68-s − 1.42·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.84302461421543, −15.31381176920941, −14.75314368647550, −14.38462134452253, −13.62775640675726, −12.89484791315154, −12.64286692605717, −11.88066725082306, −11.28733373106083, −10.88763394895300, −10.13058594554830, −9.877674936836925, −9.056051541472817, −8.598099189241422, −7.975851769893297, −7.553072478348695, −6.792266024678303, −6.057664699983795, −5.822821957924692, −4.790085431556173, −4.109224447076309, −3.373273040715950, −2.631593541674549, −1.798662532364987, −1.056633963179857, 0,
1.056633963179857, 1.798662532364987, 2.631593541674549, 3.373273040715950, 4.109224447076309, 4.790085431556173, 5.822821957924692, 6.057664699983795, 6.792266024678303, 7.553072478348695, 7.975851769893297, 8.598099189241422, 9.056051541472817, 9.877674936836925, 10.13058594554830, 10.88763394895300, 11.28733373106083, 11.88066725082306, 12.64286692605717, 12.89484791315154, 13.62775640675726, 14.38462134452253, 14.75314368647550, 15.31381176920941, 15.84302461421543
